Cobordism versions of the Hirzebruch functional equation for the virtual signature (Q2565081)

Let \(M^n\) be an oriented manifold. Any class of \(H_{n-2}(M,Z)\) can be realized uniquely up to cobordism by an oriented closed submanifold of \(M\). The submanifold intersection gives rise to the mapping \( \rho_k^{or} : (H_{n-2}(M))^k \to \Omega_{n-2}\). Composing \(\rho_k^{or}\) with the signature homomorphism \(\Omega_{n-2} \to Z\) one obtains the virtual signature \(\tau_k : (H_{n-2}(M))^k \to Z\). For \(\tau_k\) there holds the functional equation due to Hirzebruch \[ \tau_k(x_1, \dots, x_{k-1}, x + y) = \tau_k(x_1, \dots, x_{k-1}, x) + \tau_k(x_1, \dots, x_{k-1}, y) - \tau_{k+2}(x_1, \dots, x_{k-1},x, y, x+y). \] In his original proof Hirzebruch uses his well-known signature formula which expresses the signature of a manifold via its \(L\)-genus and is based on Thom's calculation of the ring \(\Omega_* \otimes Q\). Hirzebruch and Thom raised a question about a direct geometrical explanation of the functional equation [\textit{F. Hirzebruch}, Ann. Math., II. Ser. 60, 213-236 (1954; Zbl 0056.16803)]. In the paper under review the following theorem is proved. Let \(M^n\) be an oriented manifold, and let \(x, y \in H_{n-2}(M,Z)\), \(w = x + y\). Let the classes \(x\), \(y\), \(w\) be realized by closed oriented submanifolds \(X\), \(Y\), \(W\) which intersect transversally; let \(T^{n-6} = X \cap Y \cap W\). Then there exists a fiber bundle \(T \widetilde\times CP^2\) over \(T\) associated with a \(C^3\)-vector bundle \(\omega\), so that for the oriented cobordism classes holds \[ [W] = [X] + [Y] - [T \widetilde\times CP^2]. \] The proof of the theorem is based on an explicit construction of \(W\) and the cobordism is checked directly, by using cut-and-paste technique. This theorem easily implies the Hirzebruch functional equation, therefore the author writes that his construction gives a satisfactory answer to the question of Hirzebruch and Thom. Also, a similar result is proved for nonoriented cobordisms.